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8

The Uncanny Precision of the Spectral Action

Ali H. Chamseddine1,3 , Alain Connes2,3,4

1Physics Department, American University of Beirut, Lebanon2College de France, 3 rue Ulm, F75005, Paris, France

3I.H.E.S. F-91440 Bures-sur-Yvette, France4Department of Mathematics, Vanderbilt University, Nashville, TN 37240 USA

Abstract

Noncommutative geometry has been slowly emerging as a new paradigm ofgeometry which starts from quantum mechanics. One of its key features isthat the new geometry is spectral in agreement with the physical way ofmeasuring distances. In this paper we present a detailed introduction withan overview on the study of the quantum nature of space-time using thetools of noncommutative geometry. In particular we examine the suitabilityof using the spectral action as action functional for the theory. To demon-strate how the spectral action encodes the dynamics of gravity we examinethe accuracy of the approximation of the spectral action by its asymptoticexpansion in the case of the round sphere S3. We find that the two termscorresponding to the cosmological constant and the scalar curvature termalready give the full result with remarkable accuracy. This is then appliedto the physically relevant case of S3 × S1 where we show that the spectralaction in this case is also given, for any test function, by the sum of twoterms up to an astronomically small correction, and in particular all higherorder terms a2n vanish. This result is confirmed by evaluating the spectralaction using the heat kernel expansion where we check that the higher orderterms a4 and a6 both vanish due to remarkable cancelations. We also showthat the Higgs potential appears as an exact perturbation when the testfunction used is a smooth cutoff function.

1

1. An overview

Our experimental information on the nature of space-time is based on twosources:

• High energy physics based on cosmic ray information and particleaccelerator experiments, whose results are encapsulated in the Stan-dard Model of particle physics.

• Cosmology based on astronomical observations.

The large scale global picture is well described in terms of Riemannian ge-ometry and general relativity, but this picture breaks down at high energywhere the quantum effects take over. It is thus natural to look for a par-adigm of geometry which starts from the quantum framework, where therole of real variables is played by self-adjoint operators in Hilbert space.Such a framework for geometry has been slowly emerging under the nameof noncommutative geometry. One of its key features, besides the abilityto handle spaces for which coordinates no longer commute with each other,is that this new geometry is spectral. This is in agreement with physics inwhich most of the data we have, either about the far distant parts of theuniverse or about high energy physics, are also of spectral nature. The redshifted spectra of distant galaxies or the momentum eigenstates of outgo-ing particles in high energy experiments both point towards a prevalence ofspectral information. In the same vein the existing unit of time (length) isalso of spectral nature. From the mathematical standpoint it takes somedoing to obtain a purely spectral (Hilbert space theoretical) counterpart ofRiemannian geometry. One reason for the difficulty of this task is that, asis well known since the examples of J. Milnor [1], non-isometric Riemannianspaces exist which have the same spectra (for the Dirac or Laplacian oper-ators). Another reason is that the conditions for a (compact) space to be asmooth manifold are given in terms of the local charts, whose existence andcompatibility is assumed, but whose intrinsic meaning is more elusive.The paradigm of noncommutative geometry is that of spectral triple. Asits name indicates it is of spectral nature. By definition a spectral triple isa unitary Hilbert space representation of “something”. This something isan equipment that allows one to manipulate algebraically coordinates andto measure distances. The algebra of the coordinates is denoted by A andis an involutive algebra, with involution a 7→ a∗. The equipment neededto measure distances is the inverse line element D which is unbounded andfulfills D = D∗. Altogether these data fulfill some algebraic relations, e.g.if we talk about the simplest geometric space i.e. the circle S1 the relationbetween the complex unitary coordinate U and the inverse line element Dis just [D,U ] = U , which is in the vein of the Heisenberg commutationrelations.Thus, a geometry is given as a Hilbert space representation of the pair (A,D)and can be encoded by the spectral triple (A,H,D) where H is the Hilbertspace in which both the algebra A and the inverse line element D are now

2

concretely represented, the latter as an unbounded self-adjoint operator.This picture shares with the Wigner paradigm for a particle as an (irre-ducible) representation of the Poincare group the feature that it separatesthe kinematical relations from the choice of the Hilbert space representation.It is only when the latter is chosen that actual measurements of distancesbetween points x and y can be performed by formulas such as

Distance (x, y) = sup |f(x)− f(y)| , f ∈ A , ‖[D, f ]‖ ≤ 1

where indeed the norm ‖[D, f ]‖ is the operator norm in Hilbert space anddepends on the specific choice of the representation.We now have at our disposal a reconstruction theorem (cf [2]) which showsthat ordinary Riemannian spaces are neatly characterized among spectraltriples by the following kinematical relations:

• The algebra A is commutative.• The commutator [[D, a], b] = 0 for any a, b ∈ A.• The following “Heisenberg type” relation1 holds2 , for some aαj ∈ A:

(1)∑

α

aα0 [[D, aα1 ], [D, aα2 ], . . . , [D, aαn ]] = 1

together with the following spectral requirements:

• The k-th characteristic value of the resolvent of D is O(k−1/n).• Regularity.• Absolute continuity.

We refer to [2] for the precise statement. The meaning of (1) is that thedeterminant of the metric gµν does not vanish, and more precisely that itssquare root multiplied by the volume form

∑

α aα0 da

α1 ∧daα2 ∧· · · daαn gives 1.

The reason for the last two spectral requirements is technical and allows oneto specify the regularity (C∞, real analytic...) of the space and to controlthe spectral measures. The first of the spectral requirements is crucial inthat it bounds the “effective dimension” of the spectrum of the space in therepresentation. There are good physics reasons to consider that the appar-ent dimension, equal to four, of space-time is governed by the asymptoticbehavior of the eigenvalues of the line element, which is the Euclidean prop-agator. Moreover this spectral dimension is not restricted to be an integera priori and can model fractal dimension easily. The above reconstructionTheorem shows furthermore that the operator D in the spectral triple isa Dirac type operator, i.e. an order one operator with symbol given by arepresentation of the Clifford algebra. The restriction to spin manifolds isobtained by requiring a real structure i.e. an antilinear unitary operator J

1Here the multiple commutator is defined as

[T1, T2, . . . , Tn] =X

σ

ǫ(σ)Tσ(1)Tσ(2) · · ·Tσ(n)

2We assume for simplicity that the dimension n is odd

3

acting in H which plays the same role and has the same algebraic proper-ties as the charge conjugation operator in physics. When the dimension ninvolved in the reconstruction Theorem is even (rather than odd) the righthand side of (1) is now replaced by the chirality operator γ which is just aZ/2-grading in mathematical terms. It fulfills the rules

(2) γ2 = 1 , [γ, a] = 0, a ∈ A

The following further relations hold for D,J and γ

(3) J2 = ε , DJ = ε′JD, J γ = ε′′γJ, Dγ = −γD

where ε, ε′, ε′′ ∈ −1, 1. The values of the three signs ε, ε′, ε′′ depend only,in the classical case of spin manifolds, upon the value of the dimension nmodulo 8 and are given in the following table [3]:

n 0 1 2 3 4 5 6 7

ε 1 1 -1 -1 -1 -1 1 1ε′ 1 -1 1 1 1 -1 1 1ε′′ 1 -1 1 -1

In the classical case of spin manifolds there is thus a relation between themetric (or spectral) dimension given by the rate of growth of the spectrumof D and the integer modulo 8 which appears in the above table. For moregeneral spaces however the two notions of dimension (the dimension modulo8 is called the KO-dimension because of its origin in K-theory) becomeindependent since there are spaces F of metric dimension 0 but of arbitraryKO-dimension. More precisely, starting with an ordinary spin geometry Mof dimension n and taking the product M × F , one obtains a space whosemetric dimension is still n but whose KO-dimension is the sum of n withthe KO-dimension of F , which as explained can take any value modulo 8.Thus, one now has the freedom to shift the KO-dimension at very littleexpense i.e. in a way which does not alter the plain metric dimension. Asit turns out the Standard Model with neutrino mixing favors the shift ofdimension from the 4 of our familiar space-time picture to 10 = 4 + 6 = 2modulo 8 [4], [5]. The shift from 4 to 10 is a recurrent idea in string theorycompactifications, where the 6 is the dimension of the Calabi-Yau manifoldused to “compactify”. Effectively the dimension 10 is related to the existenceof Majorana-Weyl fermions. The difference between this approach and oursis that, in the string compactifications, the metric dimension of the fullspace-time is now 10 which can only be reconciled with what we experienceby requiring that the Calabi-Yau fiber remains unnaturally small. In orderto learn how to perform the above shift of dimension using a 0-dimensionalspace F , it is important to classify such spaces. This was done in [6], [7].There, we classified the finite spaces F of given KO-dimension. A space Fis finite when the algebra AF of coordinates on F is finite dimensional. Weno longer require that this algebra is commutative. The first key advantage

4

of dropping the commutativity can be seen in the simplest case where thefinite space F is given by

(4) A = Mk(C) , H = Mk(C) , D = 0 , J ξ = ξ∗, ξ ∈ HF

where the algebra A = Mk(C) is acting by left multiplication in H = Mk(C).We have shown in [8] that the study of pure gravity on the space M × Fyields Einstein gravity on M minimally coupled with Yang-Mills theory forthe gauge group SU(k). The Yang-Mills gauge potential appears as the innerpart of the metric, in the same way as the group of gauge transformations(for the gauge group SU(k)) appears as the group of inner diffeomorphisms.One can see in this Einstein-Yang-Mills example that the finite geometryfulfills a nice substitute of commutativity (of A) namely

(5) [a, b0] = 0 , ∀ a, b ∈ Awhere for any operator a in H, a0 = Ja∗J −1. This is called the orderzero condition. Moreover the representation of A and J in H is irreducible.This example is (taking γ = 1) of KO-dimension equal to 0. In [6] weclassified the irreducible (A,H, J) and found out that the solutions fall intotwo classes. Let AC be the complex linear space generated by A in L(H),the algebra of operators in H. By construction AC is a complex algebra andone only has two cases:

(1) The center Z (AC) is C, in which case AC = Mk(C) for some k.(2) The center Z (AC) is C⊕ C and AC = Mk(C)⊕Mk(C) for some k.

Moreover the knowledge of AC = Mk(C) shows that A is either Mk(C)(unitary case), Mk(R) (real case) or, when k = 2ℓ is even, Mℓ(H), whereH is the field of quaternions (symplectic case). This first case is a minorvariant of the Einstein-Yang-Mills case described above. It turns out bystudying their Z/2 gradings γ, that these cases are incompatible with KO-dimension 6 which is only possible in case (2). If one assumes that one is inthe “symplectic–unitary” case and that the grading is given by a grading ofthe vector space over H, one can show that the dimension of H which is 2k2

in case (2) is at least 2×16 while the simplest solution is given by the algebraA = M2(H) ⊕ M4(C). This is an important variant of the Einstein-Yang-Mills case because, as the center Z (AC) is C⊕ C, the product of this finitegeometry F by a manifold M appears, from the commutative standpoint,as two distinct copies of M . We showed in [6] that requiring that these twocopies of M stay a finite distance apart reduces the symmetries from thegroup SU(2) × SU(2) × SU(4) of inner automorphisms3 to the symmetriesU(1) × SU(2) × SU(3) of the Standard Model. This reduction of the gaugesymmetry occurs because of the second kinematical condition [[D, a], b] = 0which in the general case becomes:

(6) [[D, a], b0] = 0 , ∀ a, b ∈ A

3of the even part of the algebra

5

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1.0

Figure 1. Cutoff function f

Thus the noncommutative space singles out 42 = 16 as the number of phys-ical fermions, the symmetries of the standard model emerge, and moreover,as shown in [9], the model predicts the existence of right-handed neutrinos,as well as the see-saw mechanism. In the above Einstein-Yang-Mills case,the Yang-Mills fields appeared as the inner part of the metric in the sameway as the group of gauge transformations (for the gauge group SU(k)) ap-peared as the group of inner diffeomorphisms. But in that case all fieldsremained massless. It is the existence of a non-zero D for the finite space Fthat generates the Higgs fields and the masses of the Fermions and the Wand Z fields through the Higgs mechanism. The new fields are computedfrom the kinematics but the action functional, the spectral action, uses ina crucial manner the representation in Hilbert space. In order to explainthe conceptual meaning of this spectral action functional it is importantto understand in which way it encodes gravity in the commutative case.As explained above the spectrum of the Dirac operator (or similarly of theLaplacian) does not suffice to encode an ordinary Riemannian geometry.However the Einstein-Hilbert action functional, given by the integral of thescalar curvature multiplied by the volume form, appears from the heat ex-pansion of the Dirac operator. More generally it appears as the coefficientof Λ2 in the asymptotic expansion for large Λ of the trace

(7) Tr(f(D/Λ)) ∼ 2Λ4f4a0 + 2Λ2f2a2 + f0a4 + . . . + Λ−2kf−2ka4+2k + . . .

when the Riemannian geometry M is of dimension 4, and where f isa smooth even function with fast decay at infinity. The choice of thefunction f only enters in the multiplicative factors f4 =

∫∞0 f(u)u3du,

6

f2 =∫∞0 f(u)udu, f0 = f(0) and f−2k = (−1)k k!

(2k)!f(2k)(0), i.e. the deriva-

tives of even order at 0, for k ≥ 0. Thus, when f is a “cutoff” function(cf Figure 1) it has vanishing Taylor expansion at 0 and the asymptoticexpansion (7) only has three terms:

(8) Tr(f(D/Λ)) ∼ 2Λ4f4a0 + 2Λ2f2a2 + f(0)a4

The term in Λ4 is a cosmological term, the term in Λ2 is the Einstein-Hilbertaction functional, and the constant term a4 gives the integral over M of cur-vature invariants such as the square of the Weyl curvature and topologicalterms such as the Gauss-Bonnet, with numerical coefficients of order one. Itis thus natural to take the expression Tr(f(D/Λ)) as a natural spectral for-mulation of gravity. We are working in the Euclidean formulation i.e. with asignature (+,+,+,+) and the Euclidean space-time manifold is taken to becompact for simplicity. In the non-compact case we have shown in [10] howto replace the simple counting of eigenvalues of |D| of size < Λ given4 by(7), by a localized counting. This simply introduces a dilaton field. We alsotested this idea of taking the expression Tr(f(D/Λ)) as a natural spectralformulation of gravity by computing this expression in the case of manifoldswith boundary and we found [11] that it reproduces exactly the Hawking-Gibbons [12] additional boundary terms which they introduced in order torestore consistency and obtain Einstein equations as the equations of motionin the case of manifolds with boundary. Further, Ashtekar et al [13] haverecently shown that the use of the Dirac operator in a first order formalism,which is natural in the noncommutative setting, avoids the tuning and sub-traction of a constant term. One may be worried by the large cosmologicalterm Λ4f4a4 that appears in the spectral action. It is large because the valueof the cutoff scale Λ is dictated, roughly speaking, by the Planck scale sincethe term Λ2f2a2 is the gravitational action 1

16πG

∫

R√gd4x. Thus it seems

at first sight that the huge cosmological term Λ4f4a4 overrides the more sub-tle Einstein term Λ2f2a2. There is, however, and even at the classical levelto which the present discussion applies a simple manner to overcome thisdifficulty. Indeed the kinematical relation (1) in fact fixes the Riemannianvolume form to be5

(9)√gd4x =

∑

α

aα0 daα1 ∧ daα2 ∧ daα3 ∧ daα4

Thus, if we vary the metric with this constraint we are in the context ofunimodular gravity [14], and the cosmological term cancels out in the com-putation of the conditional probability of a gravitational configuration withtotal volume V held fixed. The remaining unknown, then, is the distributionof volumes dµ(V ), which is just a distribution on the half-line R+ ∋ V . Thestriking conceptual advantages of the spectral action are

4for f a cutoff function5up to a numerical factor

7

• Simplicity: when f is a cutoff function, the spectral action is justcounting the number of eigenstates of D in the range [−Λ,Λ].

• Positivity: when f ≥ 0 (which is the case for a cutoff function) theaction Tr(f(D/Λ)) ≥ 0 has the correct sign for a Euclidean action.

• Invariance: one is used to the diffeomorphism invariance of the grav-itational action but the functional Tr(f(D/Λ)) has a much strongerinvariance group, the unitary group of the Hilbert space H.

One price to pay is that, as such, the action functional Tr(f(D/Λ)) is notlocal. It only becomes so when it is replaced by the asymptotic expansion(8). This suggests that one should at least compute the next term in theasymptotic expansion (even though this term appears multiplied by thesecond derivative f ′′(0) = 0 when f is a cutoff function) just to get someidea of the size of the remainder. In fact both D and Λ have the physicaldimension of a mass, and there is no absolute scale on which they can bemeasured. The ratio D/Λ is dimensionless and the dimensionless numberthat governs the quality of the approximation (8) can be chosen to justbe the number N(Λ) of eigenvalues λ of D whose size is less than Λ, i.e.|λ| ≤ Λ. When f is a cutoff function the size of the error term in (8) shouldbe O(N−k) for any positive k, using the flatness of the Taylor expansion off at 0. In the case of interest, where M is the Euclidean space-time, a roughestimate of the size of N is the 4-dimensional volume of M in Planck unitsi.e. an order of magnitude6 of N ∼ 10214(at the present radius, see sectiontwo for details). Thus, even without the vanishing of f ′′(0), the rough error

term N−1/2 ∼ 10−107 is quite small in the approximation of the spectralaction by its local version (8). We shall in fact show that a much betterestimate holds in the simplified model of Euclidean space-time given by theproduct S3

a×S1β. Another advantage of the above spectral description of the

gravitational action is that one can now use the same action Tr(f(D/Λ))for spaces which are not Riemannian. The simplest case is the productof a Riemannian geometry M (of dimension 4) by the finite space F of(4). The only new term that appears is the Yang-Mills action functional ofthe SU(k) gauge fields which form the inner part of the metric. This newterm appears as an additional term in the coefficient a4 of Λ0, and withthe positive sign. In other words gravity on the slightly noncommutativespace M × F gives ordinary gravity minimally coupled with SU(k)-Yang-Mills gauge theory. The latter theory is massless and the fermions are inthe adjoint representation. The fermionic part of the action is easy to writesince one has the operator D whose inner fluctuations are

(10) DA = D +A+ JAJ−1 , A =∑

aj [D, bj ] , aj, bj ∈ A , A = A∗

6using the age of the universe in Planck units to estimate the spatial Euclidean direc-tions and the inverse temperature β = 1/kT also in Planck units, to set the size of theimaginary time component of the Euclidean M .

8

In the Einstein-Yang-Mills system so obtained, all fields involved are mass-less.

We now consider the product M × F of a Riemannian geometry M (ofdimension 4) by the finite space F of KO-dimension 6 which was determinedabove. The computation shows that (cf [9])

• The inner fluctuations of the metric give an U(1) × SU(2) × SU(3)gauge field and a complex Higgs doublet scalar field.

• The spectral action Tr(f(D/Λ)) plus the antisymmetric bilinear form〈Jξ,DAη〉 on chiral fermions, gives the Standard Model minimallycoupled to gravity, with the Majorana mass terms and see-saw mech-anism.

• The gauge couplings fulfill the unification constraint, the Yukawacouplings fulfill Y2 = 4g2, where Y2 is defined in eq(11), and theHiggs quartic coupling also fulfills a unification constraint.

Most of the new terms occur in the a4 term of the expansion (8). This is thecase for the minimal coupling of the Higgs field as well as its quartic self-interaction. The terms a0 and a2 get new contributions from the Majoranamasses (cf [9]), but the main new term in a2 has the form of a mass termfor the Higgs field with the coefficient −Λ2. This immediately raises thequestion of the meaning of the specific values of the couplings in the aboveaction functional. Unlike the above massless Einstein-Yang-Mills system wecan no longer take the above action simply as a classical action, would it bebecause of the unification of the three gauge couplings, which does not holdat low scale. The basic idea proposed in [8] is to consider the above actionas an effective action valid at the unification scale Λ and use the Wilsonianapproach of integrating the high frequency modes to show that one obtainsa realistic picture after “running down” from the unification scale to theenergies at which observations are done. This approach is closely relatedto the approach of Reuter [15], Dou and Percacci [16], [17]. The coarsegraining uses a much lower scale ρ which can be understood physically asthe resolution with which the system is observed. The modes with momentalarger than ρ cannot be directly observed and their effect is averaged outby the functional integral. In fact the way the renormalization group iscomputed in [16] shows that the derivative ρ∂ρΓρ of the effective action isexpressed as a trace of an operator function of the propagators and is thusof a similar nature as the spectral action itself, though the trace involvesall fields and not just the spin 1

2 fields as in the spectral action. It is anopen question to compute the renormalization group flow for the spectralaction in the context of spectral triples. One expects, as explained above,that new terms involving traces of functions of the bosonic propagator7

δ2

δDδDTr(f(D/Λ)) will be generated. The idea of taking the spectral actionas a boundary condition of the renormalization group at unification scale

7We thank John Iliopoulos for discussions on this point.

9

generates a number of severe tests. The first ones involve the dimensionlesscouplings. These include

(1) The three gauge couplings(2) The Yukawa couplings(3) The Higgs quartic coupling

As is well known, the gauge couplings do not unify in the Standard Modelbut the meeting of g2 and g3 specifies a “unification” scale of ∼ 1017 GeV.For the Yukawa couplings the boundary condition gives

(11) Y2 = 4 g2, Y2 =∑

σ

(yσν )2 + (yσe )

2 + 3 (yσu)2 + 3 (yσd )

2.

This yields a value of the top mass which is 1.04 times the observed valuewhen neglecting8 the Yukawa couplings of the bottom quarks etc...and ishence compatible with experiment. The Higgs quartic coupling (scatteringparameter) has the boundary condition of the form:

λ(Λ) = g23b

a2∼ g23

The numerical solution to the RG equations with the boundary value λ0 =0.356 at Λ = 1017 GeV gives λ(MZ) ∼ 0.241 and a Higgs mass of the order of170 GeV. This value now seems to be ruled out experimentally but this mightsimply be a clear indication of the presence of some new physics, instead ofthe “big desert” which is assumed here in the huge range of energies between102 GeV and 1017 GeV. To be more precise the above “prediction” of theHiggs mass is in perfect agreement with the one of the Standard Model,when one assumes the “big desert” (cf [19]). In a forthcoming paper [18]we show that the choice of the spectral function f could play an importantrole, even when it varies slightly from the cutoff function. This is relatedto the fact that the vev of the Higgs field is proportional to the scale Λand thus higher order corrections do contribute. This will cause the relationbetween the gauge coupling constants to be modified and to change theHiggs potential. Such gravitational corrections are known to cause sizablechanges to the Higgs mass [20].The next tests involve the dimensionful couplings. These include

(1) The inverse Newton constant Zg = 1/G.(2) The mass term of the Higgs.(3) The Majorana mass terms.(4) The cosmological constant.

Since our action functional combines gravity and the Standard Model, theanalysis of [16] applies, and the running of the couplings Z which havethe physical dimension of the square of a mass is well approximated byβZ = a1k

2 where the parameter k is fixing the cutoff scale but is considereditself as one of the couplings, while the coefficient a1 is a dimensionless

8See [9] for the precise satement.

10

number of order one. For the inverse Zg of the Newton constant, one getsthe solution:

(12) Zg = Zg(1 +1

2a1

k2

Zg)

which behaves like a constant and shows that the change in Zg is moderatebetween the low energy value Zg at k = 0 and its value at k = mP the Planck

scale, for which k2

Zg= 1. We have shown in [9] that a relation between the

moments of the cutoff function f involved in the spectral action, of the formf2 ∼ 5f0 suffices to give a realistic value of the Newton constant, providedone applies the spectral action at the unification scale Λ ∼ 1017 GeV. Theabove discussion of the running of Zg shows that this yields a reasonablelow energy value of the Newton constant G.The form βZ = a1k

2 of the running of a coupling with mass2 dimensionimplies that, as a rule, even if this coupling happens to be small at lowscale, it will necessarily be of the order of Λ2 at unification scale. For theMajorana mass terms, we explained in [9] why they are of the order ofΛ2 at unification and their role in the see-saw mechanism shows that oneshould not expect them to be small at small scale, thus a running like (12)is realistic. Things are quite different for the mass term of the Higgs. Thespectral action delivers a huge mass term of the form −Λ2H2 and one cancheck that it is consistent with the sign and order of magnitude of the qua-dratic divergence of the self-energy of this scalar field. However though thisshows compatibility with a small low energy value it does by no means al-low one to justify such a small value. Giving the term −Λ2H2 at unificationscale and hoping to get a small value when running the theory down to lowenergies by applying the renormalization group, one is facing a huge finetuning problem. Thus one should rather try to find a physical principle toexplain why one obtains such a small value at low scale. In the noncommu-tative geometry model M × F of space-time the size of the finite space Fis governed by the inverse of the Higgs mass. Thus the above problem hasa simple geometric interpretation: Why is the space F so large9 in Planck

units? There is a striking similarity between this problem and the problemof the large size of space in Planck units. This suggests that it would bevery worthwhile to develop cosmology in the context of the noncommutativegeometry model of space-time, with in particular the preliminary step of theLorentzian formulation of the spectral action.This also brings us to the important role played by the dilaton field whichdetermines the scale Λ in the theory. The spectral action is taken to be afunction of the twisted Dirac operator so that D2 is replaced with e−φD2e−φ.In [10] we have shown that the spectral action is scale invariant, except forthe dilaton kinetic energy. Moreover, one can show that after rescaling the

9by a factor of 1016.

11

physical fields, the scalar potential of the theory will be independent of thedilaton at the classical level. At the quantum level, the dilaton acquiresa Coleman-Weinberg potential [21] and will have a vev of the order of thePlanck mass [22]. The fact that the Higgs mass is damped by a factor ofe−2φ, can be the basis of an explanation of the hierarchy problem.In this paper we investigate the accuracy of the approximation of the spec-tral action by the first terms of its asymptotic expansion. We consider theconcrete example given by the four-dimensional geometry S3

a ×S1β where S3

a

is the round sphere of radius a as a model of space, while S1β is a circle of ra-

dius β viewed as a model of imaginary periodic time at inverse temperatureβ. We compute directly the spectral action and compare it with the sum ofthe first terms of the asymptotic expansion. In section two we start with theround sphere S3

a and use the known spectrum of the Dirac operator togetherwith the Poisson summation formula, to estimate the remainder when usinga smooth test function. This is then applied to the four-dimensional spaceS3a×S1

β where it is shown that, for natural test functions, the spectral actionis completely determined by the first two terms, with an error of the order

of 10−σ2where σ is the inner diameter Λµ, µ = inf(a, β) in units of the

cutoff Λ. Thus for instance an inner diameter of 10 in cutoff units yields theaccuracy of the first hundred decimal places, while an inner diameter of 1031

corresponding to the visible universe at inverse temperature of 3 Kelvin anda cutoff at Planck scale10, yields an astronomical precision of 1062 accuratedecimal places. This is then extended in the presence of Higgs fields. Theabove direct computation allows one to double check coefficients in the spec-tral action. It also implies, for S3

a × S1β, the vanishing of all the Seeley-De

Witt coefficients a2n, n ≥ 2, in the heat expansion of the square of the Diracoperator. This is confirmed in section three, by a local computation of theheat kernel expansion, where it is shown that a4 and a6 vanish due to subtlecancelations.

2. Estimate of the asymptotics

The number N(Λ) of eigenvalues of |D| which are ≤ Λ

(13) N(Λ) = # eigenvalues of D in [−Λ,Λ],

is a step function N(Λ) which jumps by the integer multiplicity of an eigen-value whenever Λ belongs to the spectrum of |D|. This integer valued func-tion is the superposition of two terms,

N(Λ) = 〈N(Λ)〉 +Nosc(Λ).

The oscillatory part Nosc(Λ) is generically the same as for a random matrix.The average part 〈N(Λ)〉 is computed by a semiclassical approximation from

10while the age of the universe in Planck units gives Λa ∼ 1061.

12

local expressions involving the familiar heat equation expansion and will nowbe carefully defined assuming an expansion of the form11

(14) Trace (e−t∆) ∼∑

aα tα (t → 0)

for the positive operator ∆ = D2. One has,

(15) ∆−s/2 =1

Γ(

s2

)

∫ ∞

0e−t∆ ts/2−1 dt

and the relation between the asymptotic expansion (14) and the ζ function,

(16) ζD(s) = Trace (∆−s/2)

is given by,

• α < 0 gives a pole at −2α for ζD with

(17) Ress=−2α ζD(s) =2 aα

Γ(−α)

• α = 0 (no log t term) gives regularity at 0 for ζD with

(18) ζD(0) = a0 .

For simple superpositions of exponentials, as Laplace transforms,

(19) f(u) =

∫ ∞

0e−su h(s) ds

we can write formally,

(20) f(t∆) =

∫ ∞

0e−st∆ h(s) ds

and

(21) Trace (f(t∆)) ∼∑

aα tα

∫ ∞

0sα h(s) ds .

For α < 0 one has,

sα =1

Γ(−α)

∫ ∞

0e−sv v−α−1 dv

and∫ ∞

0sα h(s) ds =

1

Γ(−α)

∫ ∞

0f(v) v−α−1 dv

so that

Trace (f(t∆)) ∼∑

α<0

1

2Ress=−2α ζD(s)

∫ ∞

0f(v) v−α−1 dv tα

+ ζD(0) f(0) +∑

α>0

aα tα

∫ ∞

0sα h(s) ds .(22)

11the aα defined here is equal to the Seeley-de Witt coefficients an+2α in dimension n.

13

Now we assume that the only α > 0 for which aα 6= 0 are integers and notethat

(23)

∫ ∞

0sn h(s) ds = (−1)n f (n)(0) ,

so that all the terms aα for α > 0 have vanishing coefficients when f isa cutoff function which is constant equal to 1 in a neighborhood of 0. Todefine the average part we consider the limit case f(v) = 1 for |v| ≤ 1 and0 elsewhere and get for the coefficients of (22)

(24)1

2

∫ ∞

0f(v) v−α−1 dv tα =

tα

(−2α),

which, with t = Λ−2, gives the following definition for the average part

(25) 〈N(Λ)〉 :=∑

k>0

Λk

kRess=k ζD(s) + ζD(0) .

To get familiar with this definition we shall work out its meaning in a simplecase,

Proposition 1. Assume that Spec D ⊂ Z and that the total multiplicity of

±n is P (n) for a polynomial P (x) =∑

ck xk. Then one has

〈N (Λ)〉 =∫ Λ

0P (u) du+ c , c =

∑

ck ζ(−k) ,

where ζ is the Riemann zeta function.

Proof. One has by construction, with P (x) =∑

ck xk,

ζD(s) =∑

P (n) n−s =∑

ck ζ(s− k)

Thus

Ress=k ζD(s) = ck−1

and

〈N(Λ)〉 :=∑

k>0

Λk

kck−1 + ζD(0) .

The constant ζD(0) is given by

∑

ck ζ(−k)

and is independent of Λ.

14

2.1. The sphere S4. We check the hypothesis of Proposition 1 for a roundeven sphere. We recall ([26]) that the spectrum of the Dirac operator forthe round sphere Sn of unit radius is given by

(26) Spec(D) = ±(n

2+ k) | k ∈ Z, k ≥ 0

where the multiplicity of (n2 +k) is equal to 2[n2](k+n−1

k

)

. Thus for n = 4 onegets that the spectrum consists of the relative integers, except for −1, 0, 1.The multiplicity of the eigenvalue m is 4

(

k+3k

)

for k + 2 = m which gives,for the total multiplicity of ±m

P (m) =4

3(m+ 1)m(m− 1) =

4

3(m3 −m)

which shows that one gets the correct minus sign for the scalar curvatureterm after integration using Proposition 1. Thus one gets (up to the nor-malization factor 4

3 )

(27) Tr(|D|−s) = ζ(s− 3)− ζ(s− 1)

This function has a value at s = 0 given by

ζ(−3)− ζ(−1) =1

120+

1

12=

11

120

which, taking into account the factor 43 from normalization, matches the

coefficient 11360 ×4 which appears in the spectral action in front of the Gauss-

Bonnet term, as will be shown in §3.

2.2. The sphere S3. We now want to look at the case of S3 and determinehow good the approximation of (22) is for test functions.In order to estimate the remainder of (22) in this special case we shall usethe Poisson summation formula

(28)∑

Z

h(n) =∑

Z

h(n) , h(x) =

∫

R

h(u)e−2πixudu

or rather, since the spectrum is 12 + Z in the odd case, the variant

(29)∑

Z

g(n +1

2) =

∑

Z

(−1)ng(n)

(obtained from (28) using h(u) = g(u+ 12)).

In the case of the three sphere, the eigenvalues are ±(32 + k), for k ≥ 0 with

the multiplicity 2(

k+2k

)

. Thus n + 12 has multiplicity n(n + 1). This holds

not only for n ≥ 0 but also for n ∈ Z since the multiplicity of −(n + 12)

is n(n + 1) = m(m + 1) for m = −n − 1. In particular ±12 is not in the

spectrum. Thus when we evaluate Tr(f(D/Λ)), with f an even function, weget the following sum

(30) Tr(f(D/Λ)) =∑

Z

n(n+ 1)f((n +1

2)/Λ)

15

We apply (29) with g(u) = (u2 − 14)f(u/Λ). The Fourier transform of g is

g(x) =

∫

R

g(u)e−2πixudu =

∫

R

(u2 − 1

4)f(u/Λ)e−2πixudu

= Λ3

∫

R

v2f(v)e−2πiΛxvdv − 1

4Λ

∫

R

f(v)e−2πiΛxvdv

We introduce the function f (2) which is the Fourier transform of v2f(v) andwe thus get from (29),

(31) Tr(f(D/Λ)) = Λ3∑

Z

(−1)nf (2)(Λn)− 1

4Λ∑

Z

(−1)nf(Λn)

If we take the function f in the Schwartz space S(R), then both f and f (2)

have rapid decay and we can estimate the sums

∑

n 6=0

|f(Λn)| ≤ CkΛ−k ,

∑

n 6=0

|f (2)(Λn)| ≤ CkΛ−k

which gives, for any given k, an estimate for a sphere of radius a of the form:

(32) Tr(f(D/Λ)) = (Λa)3∫

R

v2f(v)dv − 1

4(Λa)

∫

R

f(v)dv +O((Λa)−k)

The radius simply rescalesD and enters in such a way as to make the productΛa dimensionless. This can be seen by noting that the ratio D

Λ contains the

term 1Λe

µαγα∂µ and the radius enters as 1

a in the inverse dreibein eµα. Note

that, provided that k > 1 one controls the constant in front of (Λa)−k fromthe constants cj with

|xkf(x)| ≤ c1 , |xkf (2)(x)| ≤ c2 .

To get an estimate of these constants cj , say for k = 2, one can use theL1-norms of the functions ∆f(v) and ∆(v2f(v)) where ∆ = −∂2

v is theLaplacian. If we take for f a smooth cutoff function we thus get that the cjare of order one.In fact we shall soon get a much better estimate (Corollary 4 below) whichwill show that, for suitable test functions, a size ofN in cutoff units, Λa ∼ N ,

already ensures a precision of the order of e−N2. We shall work directly with

the physically more relevant model consisting of the product S3×S1 viewedas a model of the imaginary time periodic compactification of space-time ata given temperature. Our estimates will work well for a size in cutoff unitsas small as N ∼ 10 and will give the result with an astronomical precisionfor larger values. These correspond to later times since both the radius ofspace and the inverse temperature are increasing functions of time in thissimple model.

16

2.3. The product S3 × S1. We now want to move to the 4-dimensionalEuclidean case obtained by taking the product M = S3 × S1 of S3 by asmall circle. We take the product geometry of a three dimensional geometrywith Dirac operator D3 by the one dimensional circle geometry with Dirac

(33) D1 =1

βi∇θ

so that the spectrum of D1 is 1β (Z+ 1

2).

Lemma 2. Let D be the Dirac operator of the product geometry

(34) D =

(

0 D3 ⊗ 1 + i⊗D1

D3 ⊗ 1− i⊗D1 0

)

The asymptotic expansion for Λ → ∞ of the spectral action of D is given by

(35) Tr(h(D2/Λ2)) ∼ 2β ΛTr(k(D23/Λ

2)) ,

where the function k is given by

(36) k(x) =

∫ ∞

x(u− x)−1/2 h(u) du

Proof. By linearity of both sides in the function h (using the linearity ofthe transformation (36)) it is enough to prove the result for the functionh(x) = e−bx. One has

D2 =

(

D23 ⊗ 1 + 1⊗D2

1 00 D2

3 ⊗ 1 + 1⊗D21

)

and

Tr(e−bD2/Λ2) = 2Tr(e−bD2

1/Λ2)Tr(e−bD2

3/Λ2)

Moreover by (33) the spectrum of D1 is 1β (Z + 1

2) so that, using (29), and

for fixed β and b, one has for all k > 0,

Tr(e−bD21/Λ

2) ∼

√π β Λ b−1/2 +O(Λ−k) .

Thus

Tr(e−bD2/Λ2) = 2β ΛTr(

√π b−1/2 e−bD2

3/Λ2) +O(Λ−k+3)

and the equality (35) follows from∫ ∞

x(u− x)−1/2 e−bu du =

√π b−1/2 e−bx

which shows that the function k associated to h(x) = e−bx by the linear

transformation (36) is k(x) =√π b−1/2 e−bx.

One can write (36) in the form

(37) k(x) =

∫ ∞

0v−1/2 h(x+ v) dv ,

17

which shows that k has right support contained in the right support of hi.e. that if h vanishes identically on [a,∞[ so does k. It also gives a goodestimate of the derivatives of k since

∂nxk(x) =

∫ ∞

0v−1/2 ∂n

xh(x+ v) dv .

In fact, in order to estimate the size of the remainder in the asymptoticexpansion of the spectral action for the product M = S3 ×S1, we shall nowuse the two dimensional form of (29),

(38)∑

Z2

g(n +1

2,m+

1

2) =

∑

Z2

(−1)n+mg(n,m)

where the Fourier transform is given by

(39) g(x, y) =

∫

R2

g(u, v)e−2πi(xu+yv)dudv

For the operator D of (34), and taking for D3 the Dirac operator of the3-sphere S3

a of radius a, the eigenvalues of D2/Λ2 are obtained by collectingthe following

(1

2+ n)2 (Λa)−2 + (

1

2+m)2 (Λβ)−2 , n,m ∈ Z

with the multiplicity 2n(n+ 1) for each n,m ∈ Z. Thus, more precisely

Tr(h(D2/Λ2)) =∑

Z2

2n(n+ 1)h((1

2+ n)2 (Λa)−2 + (

1

2+m)2 (Λβ)−2)

which is of the form:

(40) Tr(h(D2/Λ2)) =∑

Z2

g(n+1

2,m+

1

2)

where

(41) g(u, v) = 2(u2 − 1

4)h(u2 (Λa)−2 + v2 (Λβ)−2)

One has

g(0, 0) =

∫

R2

g(u, v)dudv = 2

∫

R2

(u2 − 1

4)h(u2 (Λa)−2 + v2 (Λβ)−2)dudv

= 2 (Λa) (Λβ)

∫

R2

((Λa)2 x2 − 1

4)h(x2 + y2)dxdy

using u = x (Λa) and v = y (Λβ) . Thus we get:

(42) g(0, 0) = 2π (Λβ) (Λa)3∫ ∞

0h(ρ2)ρ3dρ− π (Λβ) (Λa)

∫ ∞

0h(ρ2)ρdρ

To estimate the remainder, given by the sum∑

(n,m)6=(0,0)

(−1)n+mg(n,m)

18

we treat separately the Fourier transforms of

g1(u, v) = u2h(u2 (Λa)−2+v2 (βΛ)−2) , g2(u, v) = h(u2 (Λa)−2+v2 (Λβ)−2)

One has

g2(n,m) =

∫

R2

g2(u, v)e−2πi(nu+mv)dudv

= Λ2βa

∫

R2

h(x2 + y2)e−2πi(nΛax+mΛβy)dxdy = Λ2βaκ2(nΛa,mΛβ)

where the function of two variables κ2(u, v) is the Fourier transform,

(43) κ2(u, v) =

∫

R2

h(x2 + y2)e−2πi(ux+vy)dxdy = κ(u2 + v2)

The function κ is related to the function k(x) defined by (36), and one has

(44) κ(u2) =

∫

R

k(x2)e−2πiuxdx

so that κ(u2) is the Fourier transform of k(x2).For g1 one has, similarly,

g1(n,m) =

∫

R2

g1(u, v)e−2πi(nu+mv)dudv

= Λ4βa3∫

R2

x2h(x2 + y2)e−2πi(nΛax+mΛβy)dxdy = Λ4βa3κ1(nΛa,mΛβ)

where the function of two variables κ1(u, v) is the Fourier transform,

κ1(u, v) =

∫

R2

x2h(x2 + y2)e−2πi(ux+vy)dxdy

which is given in terms of (43) by

(45) κ1(u, v) = −π−2(u2κ′′(u2 + v2) +1

2κ′(u2 + v2))

Now for any test function h in the Schwartz space S(R), the functionx2h(x2+y2) is in the Schwartz space S(R2) and thus we have for its Fouriertransform, and any k > 0, an estimate of the form

(46) |κ1(u, v)| ≤ Ck(u2 + v2)−k

We thus get, for k > 2,

|∑

(n,m)6=(0,0)

(−1)n+mg1(n,m)| ≤∑

(n,m)6=(0,0)

|g1(n,m)|

= Λ4βa3∑

(n,m)6=(0,0)

|κ1(nΛa,mΛβ)| ≤ CkΛ4βa3

∑

(n,m)6=(0,0)

((nΛa)2+(mΛβ)2)−k

≤ CkΛ4βa3(Λµ)−2k

∑

(n,m)6=(0,0)

(n2 +m2)−k , µ = inf(a, β)

We thus get, using a similar estimate for g2,

19

Theorem 3. Consider the product geometry S3a × S1

β. Then one has, for

any test function h in the Schwartz space S(R), the equality

(47) Tr(h(D2/Λ2)) = 2πΛ4βa3∫ ∞

0h(ρ2)ρ3dρ−πΛ2βa

∫ ∞

0h(ρ2)ρdρ+ǫ(Λ)

where ǫ(Λ) = O(Λ−k) for any k is majorized by

|ǫ(Λ)| ≤ 2Λ4βa3∑

(n,m)6=(0,0)

|κ1(nΛa,mΛβ)|+1

2Λ2βa

∑

(n,m)6=(0,0)

|κ2(nΛa,mΛβ)| .

with κj defined in (43) and (45).

This implies that all the Seeley coefficients a2n vanish for n ≥ 2, and weshall check this directly for a4 and a6 in §3.This vanishing of the Seeley coefficients does not hold for the 4 sphere and itis worth understanding why one cannot expect to use the Poisson summationin the same way for the 4 sphere. The problem when one tries to use thePoisson formula as above is that, e.g. for the heat kernel, one is dealing with

a function like |x|e−tx2which is not smooth and whose Fourier transform

does not have rapid decay at ∞.

2.4. Specific test functions. We shall now concretely evaluate the re-mainder in Theorem 3 for analytic test functions of the form

(48) h(x) = P (πx)e−πx

where P is a polynomial of degree d. The Fourier transforms of the functionsof two variables x2h(x2 + y2) and h(x2 + y2) are of the form

κj(u, v) = Pj(u, v)e−π(u2+v2)

where the Pj are polynomials. More precisely, since the Fourier transform of

e−λπ(x2+y2) is 1λe

−π (u2+v2)λ one obtains the formula for P2 by differentiation

at λ = 1 and get

κ2(u, v) = P (−∂λ)λ=11

λe−π (u2+v2)

λ

which is of the form

κ2(u, v) = Q(π(u2 + v2))e−π(u2+v2)

where Q is a polynomial of degree d. The transformation P 7→ Q = T (P ) isgiven by

(49) Q(z) = P (−∂λ)λ=11

λe−

zλ

Moreover one then gets

κ1(u, v) = −(2π)−2∂2uκ2(u, v)

= (u2Z1(π(u2 + v2)) + Z2(π(u

2 + v2)))e−π(u2+v2)

20

where

(50) Z1 = −Q+ 2Q′ −Q′′ , Z2 =1

2π(Q−Q′)

We let CP be the sum of the absolute values of the coefficients of Q = T (P ).

Corollary 4. Consider the product geometry S3a × S1

β. Let µ = inf(a, β).

Then one has, with h any test function of the form (48), the equality

(51) Tr(h(D2/Λ2)) = 2πΛ4βa3∫ ∞

0h(ρ2)ρ3dρ−πΛ2βa

∫ ∞

0h(ρ2)ρdρ+ǫ(Λ)

where, assuming µΛ ≥√

d(1 + log d) and µΛ ≥ 1,

(52) |ǫ(Λ)| ≤ Ce−π2(µΛ)2 , C = Λ4βa3CP (8 + 6d+ 2d2)

Proof. One has

xke−x/2 ≤ 1 , ∀x ≥ 3k(1 + log k)

Thus, for (n,m) 6= (0, 0) one has

|κ2(nΛa,mΛβ)| ≤ CP e−π

2((nΛa)2+(mΛβ)2)

since π((nΛa)2 + (mΛβ)2) ≥ 3d(1 + log d). Moreover, since e−π2(µΛ)2 ≤ 1

4 ,one gets

∑

(n,m)6=(0,0)

e−π2((nΛa)2+(mΛβ)2) ≤ 8e−

π2(µΛ)2

and∑

(n,m)6=(0,0)

|κ2(nΛa,mΛβ)| ≤ 8CP e−π

2(µΛ)2

A similar estimate using (50) yields∑

(n,m)6=(0,0)

|κ1(nΛa,mΛβ)| ≤ (2 + 3d+ d2)CP e−π

2(µΛ)2

Thus by Theorem 3, the inequality (52) holds for

C = CP (2Λ4βa3(2 + 3d+ d2) + 4Λ2βa) .

One then uses the hypothesis µΛ ≥ 1 to simplify C.

The meaning of Corollary 4 is that the accuracy of the asymptotic expansion

is at least of the order of e−π2(µΛ)2 . Indeed the term Λ4βa3 in (52) is the

dominant volume term in the spectral action and the other terms in theformula for C are of order one. Thus for instance for a size µΛ ∼ 100 onegets that the asymptotic expansion accurately delivers the first 6820 decimalplaces of the spectral action. Note that some test functions of the form (48)give excellent approximations to cutoff functions, in particular

(53) hn(x) =

n∑

0

(πx)k

k!e−πx

21

The graph of hn(x2) is shown in Figure 1 for n = 20. For h = h20 the

computation gives CP (8 + 6d + 2d2) ≤ 2 × 106 so that this constant onlyinterferes with the last six decimal places in the above accuracy.

In our simplified physical model we test the approximation of the spectralaction by its asymptotic expansion for the Euclidean model

E(t) = S3a(t) × S1

β(t)

where space at a given time t is given by a sphere with radius a(t) and β(t)is a uniform value of inverse temperature. One can then easily see that theabove approximation to the spectral action is fantastically accurate, goingbackwards in time all the way up to one order lower than the Planck energy.In doing so the radius a(t) varies between at least ∼ 1061 Planck units and10 Planck units (i.e. 10−34 m), while the temperature varies between 2.7Kand

(

1031)

K. It is for an inner size less than 10 in Planck units that theapproximation does break down.

Remark 5. For later purpose it is important to estimate the constant CP

in terms of the coefficients of the polynomial P . Let then P (z) = zn. Onehas h(x) = (πx)ne−πx and the function k(x) associated to h by (36) is

k(x) =

∫

R

h(x+ y2)dy = πne−πxn∑

0

(

n

k

)

xn−k

∫

R

y2ke−πy2dy

= π−1/2e−πxn∑

0

(

n

k

)

Γ(1

2+ k)(πx)n−k

To obtain Q = T (P ) one then needs to compute the Fourier transform κ(u2)

of the function k(x2) as in (44). The Fourier transform of (πx2)me−πx2is

ℓm(u) = (−4π)−m∂2mu e−πu2

= Lm(πu2)e−πu2

and one checks, using the relation

Lm+1(z) = 1/2((1 − 2z)Lm(z) + (−1 + 4z)L′m(z) − 2zL′′

m(z)

that the sign of the coefficient of zk in Lm(z) is (−1)k. Thus the sum of the

absolute values of the coefficients of Lm is equal to Lm(−1) = ℓm(iπ−1/2)/e.Thus since the above sum giving k(x) has positive coefficients we get that,

for P (z) = zn, the constant CP is given by Q(π(u2 + v2)e−π(u2+v2))/e for(u, v) = (iπ−1/2, 0), which gives

CP =

∫

R2

πn(y2 + x2)ne−πy2−πx2+2√πx−1dxdy .

One then gets

(54) CP ≤ 2

∫ ∞

0u2n+1e−(u−1)2du = O(λnn!) , ∀λ > 1 .

22

Thus, for an arbitrary polynomial P (z) =∑d

0 akzk one has

(55) CP ≤ 2

∫ ∞

0|P |(u2)e−(u−1)2udu , |P |(z) =

∑

|ak|zk

2.5. The Higgs potential. We now look at what happens if one performsthe following replacement on the operator

D2 7→ D2 +H2

where H is a constant. This amounts with the above notations to thereplacement

(56) h(u) 7→ h(u+H2/Λ2) .

As long as H2/Λ2 is of order one, we can trust the asymptotic expansionand we just need to understand the effect of this shift on the two terms of(47). We look at the first contribution, i.e.

2πΛ4βa3∫ ∞

0h(ρ2)ρ3dρ = πΛ4βa3

∫ ∞

0uh(u)du

We let x = H2/Λ2, and get, after the above replacement,∫ ∞

0uh(u+x)du =

∫ ∞

x(v−x)h(v)dv =

∫ ∞

0(v−x)h(v)dv−

∫ x

0(v−x)h(v)dv

=

∫ ∞

0vh(v)dv − x

∫ ∞

0h(v)dv −

∫ x

0(v − x)h(v)dv

The first term corresponds to the initial contribution of πΛ4βa3∫∞0 uh(u)du.

The second term gives

(57) − πΛ4βa3x

∫ ∞

0h(v)dv = −πΛ2βa3H2

∫ ∞

0h(v)dv

which is the expected Higgs mass term from the Seeley–de Witt coefficienta2. To understand the last term we assume that h is a cutoff function.

Lemma 6. If h is a smooth function constant on the interval [0, c], thenfor x = H2/Λ2 ≤ c the new terms arising from the replacement (56) are

given by

(58) − πΛ2βa3∫ ∞

0h(v)dv H2 +

1

2πβah(0) H2 +

1

2πβa3h(0) H4

Proof. For the perturbation of πΛ4βa3∫∞0 uh(u)du, besides (57), we just

need to compute the last term −∫ x0 (v − x)h(v)dv, and one has

−∫ x

0(v − x)h(v)dv = h(0)

∫ x

0(x− v)dv =

1

2h(0)x2

since h is constant on the interval [0, x].

23

We then look at the effect on the second contribution, i.e.

−πΛ2βa

∫ ∞

0h(ρ2)ρdρ = −1

2πΛ2βa

∫ ∞

0h(u)du

We let, as above, x = H2/Λ2, and get∫ ∞

0h(u+ x)du =

∫ ∞

xh(v)dv =

∫ ∞

0h(v)dv −

∫ x

0h(v)dv

Thus the perturbation, under the hypothesis of Lemma 6 is

−1

2πΛ2βa(−xh(0)) =

1

2πβah(0) H2

The three terms in formula (58) correspond to the following new terms forthe spectral action

• The Higgs mass term coming from the Seeley–de Witt coefficient a2.• The RH2 term coming from the Seeley–de Witt coefficient a4.• The Higgs potential term in H4 coming from the Seeley–de Wittcoefficient a4.

We can now state the analogue of Theorem 3 as follows

Theorem 7. Consider the product geometry S3a × S1

β. Let µ = inf(a, β).

Then one has, with h any test function of the form (48), the equality

Tr(h((D2 +H2)/Λ2)) = 2πΛ4βa3∫ ∞

0h(ρ2)ρ3dρ− πΛ2βa

∫ ∞

0h(ρ2)ρdρ

+πΛ4βa3 V (H2/Λ2) +1

2πΛ2βaW (H2/Λ2) + ǫ(Λ)

where

(59) V (x) =

∫ ∞

0u(h(u+ x)− h(u))du , W (x) =

∫ x

0h(u)du

and, assuming µΛ ≥√

d(1 + log d), µΛ ≥ 1, and H2Λ−2 ≤ c/π,

(60) |ǫ(Λ)| ≤ Ce−π2(µΛ)2 , C = Λ4βa3C ′

P (8 + 6d+ 2d2)

where, with P (z) =∑d

0 akzk one has

C ′P = 4

∫ ∞

0|P |(u2 + c)e−(u−1)2udu , |P |(z) =

∑

|ak|zk

Proof. The new terms simply express the replacement (56) in the formula of

Theorem 3. The new function h thus obtained is still of the form (48) sinceit is obtained from h by a translation. It thus only remains to estimate CP

where P is the polynomial such that h(u) = P (πu)e−πu. For P (z) = zn theconstant CP for a translation u 7→ u+ x, x ≥ 0 of the variable, is less thanthe constant CPx for the polynomial

Px(πu) = P (π(u+ x)) =∑

(

n

k

)

(πx)n−k(πu)k

24

Thus, by Remark 5, (55), the constant CPx is estimated by

CPx ≤ 2

∫ ∞

0(u2 + πx)ne−(u−1)2udu

which is an increasing function of x and thus only needs to be controlled forx = c/π in our case.

For instance, for h = h20 the computation gives C ′P (8 + 6d+2d2) ≤ 3× 107

for c = 1, so that this constant only interferes with the last seven decimalplaces in the accuracy which is the same as in Corollary 4.Moreover as shown in Lemma 6, when h is close to a true cutoff function

(61) πΛ4βa3 V (H2/Λ2) +1

2πΛ2βaW (H2/Λ2)

= −2πΛ2βa3∫ ∞

0h(ρ2)ρdρH2 +

1

2πβah(0) H2 +

1

2πβa3h(0) H4 + δ

where the remainder δ is estimated from the Taylor expansion of h at 0. Forinstance for the functions hn of (53), one has by construction 0 ≤ hn(x) ≤ 1for all x and since

hn(x) = 1−∑

a(n, k)xn+k+1 , a(n, k) = (−1)k/((n + k + 1)n!k!)

one gets, for h = hn the estimate

|δ| ≤ πΛ4βa3xn+3

(n+ 3)(n + 1)!+ πΛ2βa

xn+2

2(n + 2)!, x = H2/Λ2 .

While the function W is by construction the primitive of h, and is increasingfor h ≥ 0 one has, under the hypothesis of positivity of h,

Lemma 8. The function V (x) is decreasing with derivative given by

V ′(x) = −∫ ∞

xh(v)dv

The second derivative of V (x) is equal to h(x).

Proof. One has

V ′(x) =

∫ ∞

0uh′(u+ x)du = [uh(u+ x)]∞0 −

∫ ∞

0h(u+ x)du

which gives the required results.

3. Seeley–De Witt coefficients and Spectral Action on S3 × S1

In this section we shall compute the asymptotic expansion of the spectralaction on the background geometry of S3 × S1 using heat kernel methods.This will enable us to check independently the accuracy of the estimatesderived in the last section. This background is physically relevant since itcan be connected with simple cosmological models. We refer to [23], [8] forthe formulas and the method of the computation. The general method we

25

use is also explained in great detail in a forthcoming paper [18]. We startby computing a0 :

a0 =Tr(1)

16π2

∫ √gd4x =

1

4π2

∫

S3

√

3gd3x

∫

S1

dx

=1

4π2

(

2π2a3)

(2πβ) = πβa3

where β is the radius of S1β and the volume of the three sphere S3

a of radius

a is 2π2a3 [25] .Next we calculate a2

a2 =1

16π2

∫

d4x√gTr

(

E +1

6R

)

where E is defined from the relation

D2 = − (gµν∇µ∇ν + E)

where for pure gravity we have

E = −1

4R

so that (using Tr(1) = 4 )

a2 =1

4π2

(

−R

12

)∫

d4x√g

since the curvature is constant. The curvature tensor12 is, using the coordi-nates of [25] for the three sphere S3

a with labels i, j, k, l and the label 4 forthe coordinate in S1

β,

Rijkl = −a−2 (gikgjl − gilgjk) , i, j, k, l = 1, · · · 3Rijk4 = 0

Ri4j4 = 0

where gij is the metric on the three sphere as in [25]. The Ricci tensor isgiven, following the sign convention of [24] which introduces a minus sign inpassing from the curvature tensor to the Ricci tensor, by

Rij = −gklRikjl = 2a−2gij

Ri4 = 0

R44 = 0

Thus the scalar curvature is

R = gijRij =6

a2

12the sign convention for this tensor is the same as in [23]

26

and the a2 term in the heat expansion simplifies to

a2 = −πβa

(

1

2

)

Next for a4 we have

a4 =1

16π2

1

360

∫

M

d4x√g Tr

(

12R µ;µ + 5R2 − 2RµνR

µν

+2RµνρσRµνρσ + 60RE + 180E2 + 60E µ

;µ + 30ΩµνΩµν)

where for the pure gravitational theory, we have

E = −1

4R, Ωµν =

1

4R ab

µν γab

In this case it was shown in [8] that a4 reduces to

a4 =1

4π2

1

360

∫

d4x√g(

5R2 − 8R2µν − 7R2

µνρσ

)

=1

4π2

1

360

∫

d4x√g(

−18C2µνρσ + 11R∗R∗)(62)

which is obviously scale invariant. The Weyl tensor Cµνρσ is defined by

Cµνρσ = Rµνρσ +1

2(Rµρgνσ −Rνρgµσ −Rµσgνρ +Rνσgµρ)

− 1

6(gµρgνσ − gνρgµσ)R

This tensor vanishes on S3×S1 as can be seen by evaluating the components

Cijkl = a−2 [−(gikgjl − gilgjk) + 2(gikgjl − gilgjk)− (gikgjl − gilgjk)] = 0

Cijk4 = 0

Ci4k4 = 0

Similarly the Gauss-Bonnet term

R∗R∗ =1

4ǫµνρσǫαβγδR

αβµν R γδ

ρσ

= ǫijk4ǫαβγδ

(

R αβij R γδ

k4

)

= 0

The next step of calculating a6 is in general extremely complicated, butfor spaces of constant curvature the expression simplifies as all covariantderivatives of the curvature tensor, Riemann tensor and scalar curvaturevanish. The non-vanishing terms are, using Theorem 4.8.16 of [23] and the

27

above sign convention for the Ricci tensor Rµν and the scalar curvature,

a6 =1

16π2

∫

d4x√gTr

(

1

9 · 7!(

35R3 − 42RR2µν + 42RR2

µνρσ

− 208RµνRµρRνρ − 192RµρRνσRµνρσ − 48RµνRµρσκRνρσκ

−44RµνρσRµνκλRρσκλ − 80RµνρσRµκρλRνκσλ)

+1

360

(

−12ΩµνΩνρΩρµ − 6RµνρσΩµνΩρσ − 4RµνΩµρΩνρ + 5RΩ2µν

+60E3 + 30EΩ2µν + 30RE2 + 5R2E − 2R2

µνE + 2R2µνρσE

))

We can now compute each of the above eighteen terms. These are listed inan appendix. Collecting these terms we obtain that the integrand is

− 4a−6

9 · 7!(

−35 · 63 + 42 · 72− 42 · 72 + 208 · 24− 192 · 24 + 48 · 24− 44 · 24− 80 · 6)

− 4a−6

360

(

9 + 18− 12 + 45 +15 · 27

2− 5 · 27

2− 15 · 27 + 10 · 27− 36 + 36

)

= a−6

(

2

3− 2

3

)

= 0

implying that

a6 = 0 ,

which shows that the cancelation is highly non-trivial. We conclude thatthe spectral action, up to terms of order 1

Λ4 is given by

S = Λ4

∫ ∞

0xh (x) dx

(

πβa3)

− Λ2

∫ ∞

0h (x) dx

(

πβa1

2

)

+O(

Λ−4)

After making the change of variables x = ρ2 we get

S = (πβΛ)

[

2 (Λa)3∫ ∞

0ρ3h

(

ρ2)

dρ− (Λa)

∫ ∞

0ρh

(

ρ2)

dρ

]

+O(

Λ−4)

This confirms equation (47) and shows that, to a very high degree of accu-racy, the spectral action on S3 × S1 is given by the first two terms.

Remark 9. It is worth noting that one can also check the value of theGauss-Bonnet term on S4 and show that it agrees with the value obtainedin (27). To see this note that the Riemann tensor in this case is given by([25])

Rµνρσ = −a−2 (gµρgνσ − gµσgνρ)

which implies13 that

Cµνρσ = 0

R∗R∗ = 6a−4

13One can double check the value of R∗R∗ using the Gauss–Bonnet Theorem.

28

and thus

a4 =1

4π2

11

60a−4

∫

S4

d4x√g

The volume of S4 is

V4 =

∫

S4

d4x√g =

2π52

Γ(

52

)a4 =8π2

3a4

and this implies that

a4 =11

360× 4

which agrees exactly with the calculation of (27) based on zeta functions.

Appendix

In this appendix we compute the eighteen non-vanishing terms that appearin the a6 term of the heat kernel expansion. Using the properties

R2µν = R2

ij = 12a−4

R2µνρσ = R2

ijkl = 12a−4

35R3 = 35(6)3a−6

−42RR2µν = −42 (6) (12) a−6

−208RµνRµρRνρ = −208 (2)3 gijgikgjk = 208 (2)3 (3) a−6

−192RµρRνσRµνρσ = −192RikRjlRijkl

= 192 (2)2 gikgjl (gikgjl − gilgjk) a−6

= 192 (24) a−6

−48RµνRµρσκRνρσκ = −48 (2) gij (gikglm − gilgkm) (gjkglm − gjlgkm) a−6

= −48 (4) gij (2gij) a−6 = −48 (24) a−6

−44RµνρσRµνκλRρσκλ = 44 (gikgjl − gilgjk) (gipgjq − giqgjp) (gkpglq − glqglp) a−6

= 44 (4) (6) a−6

−80RµνρσRµκρλRνκσλ = 80 (gikgjl − gilgjk) (gikgpq − giqgpk) (gjlgpq − gjqgpl) a−6

= 80 (3gjlgpq − gpqgjl − gljgpq + glqgjp) (gjlgpq − gjqgpl) a−6

= 80 (9− 3 + 3− 3) a−6

= 80 (6) a−6

Collecting the first set of terms we get

− 4a−6

9 · 7!(

−35 · 63 + 42 · 72− 42 · 72 + 208 · 24− 192 · 24 + 48 · 24− 44 · 24− 80 · 6)

=2

3a−6

29

Now we continue with the second set of terms

−12Tr (ΩµνΩνρΩρµ) = −12

(

1

4

)3

Tr (γabγcdγef )Rab

µν R cdνρ R ef

ρµ

= Tr (1) 12

(

1

4

)3

(8)RµνabRνρbcRρµac

= −3

2(gikgjl − gilgjk) (gjlgpq − gjpglq) (gpkgiq − gpigkq) a

−6Tr (1)

= −3

2(3gikgpq − gikgpq − gikgpq + giqgpk) (gpkgiq − gpigkq) 4a

−6

= −3

2(3− 3 + 9− 3) 4a−6

= −9 · 4a−6

−6RµνρσTr (ΩµνΩρσ) = − 6

42RµνρσTr (γabγcd)R

abµν R cd

ρσ

=12

16RµνρσR

abµν RρσabTr (1)

= −3

4(gikgjl − gilgjk) (gipgjq − giqgjp) (gkpglq − gkqglp) 4a

−6

= −3 (9− 3) 4a−6 = −18 · 4a−6

−4RµνTr (ΩµρΩνρ) = −1

4RµνTr (γabγcd)R

abµρ R cd

νρ

=1

2RµνR

abµρ RνρabTr (1)

=1

2(2) gij (gipgmq − giqgmp) (gjpgmq − gjqgmp) 4a

−6

= 2 (9− 3) 4a−6 = 12 · 4a−6

5RTr(

Ω2µν

)

=5

16RTr (γabγcd)R

abµν R cd

µν

= −5

8RR2

µνρσTr (1)

= −5

8(6) (12) 4a−6

= −45 · 4a−6

30

60Tr(

E3)

= 60

(

−1

4

)3

R3Tr (1)

= 60

(

−1

4

)3

(6)3 · 4a−6

= −1

2(15 · 27) · 4a−6

30Tr(

EΩ2µν

)

= 30

(

−3

2

)

(−2)

(

−1

4

)2

a−2R2µνρσTr (1)

=90

16(12)4a−6

=1

2(5 · 27) · 4a−6

30RE2Tr (1) =30

16R3Tr (1)

=

(

15

8

)

(6)3 4a−6

= (15 · 27) · 4a−6

5R2ETr (1) = −5

4R3 · 4

= −5

4(6)3 4a−6

= − (10 · 27) · 4a−6

−2R2µνETr (1) = −2R2

µν

(

−R

4

)

4

= −2 (12)

(

−3

2

)

4a−6

= 36 · 4a−6

2R2µνρσETr (1) = 2 (12)

(

−3

2

)

4a−6

= −36 · 4a−6

Collecting the second set of terms we get

−4a−6

360

(

9 + 18− 12 + 45 +15 · 27

2− 5 · 27

2− 15 · 27 + 10 · 27− 36 + 36

)

= −2

3a−6

Thus the sum of all the terms in a6 is zero.

31

Acknowledgements

The research of A. H. C. is supported in part by the Arab Fund for Socialand Economic Development.

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E-mail address: chams@aub.edu.lbE-mail address: alain@connes.org